Of course everyone is indeed entitled to their opinion. Yet, the phrase “I’m entitled to my opinion” has a negative undertone.
It gives the impression that you want to close the conversation and do not care to hear the other opinion(s) being offered in the discussion. Even worse, it has the connotation that further discussion is not acceptable and that to continue would be impolite and inappropriate. It says “I will not budge even one iota toward your point of view,” or “I don’t wish to hear the evidence that contradicts my opinion.”
Think about when you have heard (or said) this phrase. Often it is when someone has been cornered and does not want to pursue the conversation further. It comes across as defensive and childlike.
A better way might be to say “That does sound interesting, I will think about it,” or “I can see both perspectives on this.” Whether you end up thinking about it is up to you, but you have not slammed the door on the conversation.
Similarly, “You are entitled to your own opinion” also shuts down conversations. The speaker is insinuating that the other person is misguided and that they are taking the moral high road by stepping away from the argument.